WASHINGTON (Reuters) - Republican Senator Lindsey Graham said on Sunday that chances for a limited "fiscal cliff" deal in the next 48 hours were "exceedingly good," as talks to avoid tax increases and spending cuts on New Year's Day went down to the wire.

"I think people don't want to go over the cliff if we can avoid it," Graham, a leading conservative, said on Fox News Sunday.

He predicted that Republicans in Congress were going to accept some form of tax increases for wealthier Americans and this would hand a victory to President Barack Obama, although a leading Democrat warned of a big gap in talks between the parties.

Obama was "going to get tax-rate increases" on upper income Americans in the talks as he wants, Graham said.

The leading Senators from both parties are working on a stopgap fiscal cliff deal to avoid taxes increasing for almost all Americans on New Year's Day. They would probably leave the details of some other thorny fiscal issues, like government spending cuts, until January.

Graham urged Republicans to make a stand later when the time comes for Congress to decide on raising the debt ceiling.

"This deal won't affect the debt situation, it will be a political victory for the president and I hope we'll have the courage of our convictions when it comes time to raise the debt ceiling to fight for what we believe as Republicans, but hats off to the president, he won," Graham said.

Lawmakers are seeking a last-minute deal that would set aside $600 billion in tax increases and across-the-board government spending cuts that are set to start within days. If Congress does not act, the measures would likely push the United States into a recession.

Senator Richard Durbin, a member of the Democratic leadership, told the "Face the Nation" show on CBS that there was still a "chasm" between the two parties in fiscal cliff negotiations, but anything was possible when Congress faced a deadline.

Graham said that if the Senate was unable to come up with the votes, it would be more difficult for any deal to get through the House of Representatives.

"And I want to vote for it, even though I won't like it ... because the country has got a lot at stake here," he said. (Reporting By Tabassum Zakaria; Editing by David Brunnstrom)

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